Title：Hopf algebras and homotopy Lie algebras attached to closed embedding of schemes

Abstract：Let $Z\subseteq X$ be a closed subscheme of a scheme $X$ with a defining ideal sheaf $ \mathcal I\subseteq \mathcal O_X$. The structure sheaf $ \mathcal O_Z$ is a sheaf of $ \mathcal O_X$-modules on $X$ supported on $Z$. The graded sheaves $\mathcal Tor_{*}^{\mathcal O_X}(\mathcal O_Z,\mathcal O_Z)$  and $\mathcal Ext^{*}_{\mathcal O_X}(\mathcal O_Z,\mathcal O_Z$ on $X$ are supported on $Z$ and are quasi-coherent sheaves of $ \mathcal O_Z$-modules. It turns out that $\mathcal Tor_{*}^{\mathcal O_X}(\mathcal O_Z,\mathcal O_Z)$ and $\mathcal Ext^{*}_{\mathcal O_X}(\mathcal O_Z,\mathcal O_Z)$ have Hopf algebra structures (when $X$ is a Noetherian scheme).
   In this talk I will explain how a Koszul-Tate resolution of $ \mathcal O_Z$ in the category of locally free $\mathcal O_X$-modules as a  commutative and cocommutative dg Hopf algebras with divided power structures which gives rise to the Hopf algebra structure on $\mathcal Tor$ and $\mathcal Ext$. In this case there is a sheaf of graded Lie algebras $\mathcal G$ over $\mathcal O_Z$ such that $\mathcal Ext=U(\mathcal G)$. This Lie algebra is called the Homotopy Lie algebras constructed for H-spaces by Milnor-Moore in algebraic topology. This Lie algebra plays the role of tangent complexes. When $Z$ is a complete intersection, this Lie algebra corresponds to obstruction theory in complex algebraic geometry.

Speaker：Zongzhu Lin, Kansas State University, Professor. Lin works on representation theory，his results are published in Invent. Math., Adv. Math, Comm. Math. Phys, etc.

Date：9:30am-10:30am 2025-6-24（Tuesday）.

Venue: 617

Organizer：School of Mathematical Science

Inviter：Libin Li, Zhiqiang Yu

Students and teachers who are interested in Tensor categories are welcome.